Phys­ic­ally-based dif­fer­en­ti­able ren­der­ing has re­cently emerged as an at­tract­ive new tech­nique for solv­ing in­verse prob­lems that re­cov­er com­plete 3D scene rep­res­ent­a­tions from im­ages. The in­ver­sion of shape para­met­ers is of par­tic­u­lar in­terest but also poses severe chal­lenges: shapes are in­ter­twined with vis­ib­il­ity, whose dis­con­tinu­ous nature in­tro­duces severe bi­as in com­puted de­riv­at­ives un­less costly pre­cau­tions are taken. Shape rep­res­ent­a­tions like tri­angle meshes suf­fer from ad­di­tion­al dif­fi­culties, since the con­tinu­ous op­tim­iz­a­tion of mesh para­met­ers can­not in­tro­duce to­po­lo­gic­al changes. One com­mon solu­tion to these dif­fi­culties en­tails rep­res­ent­ing shapes us­ing signed dis­tance func­tions (SD­Fs) and gradu­ally ad­apt­ing their zero level set dur­ing op­tim­iz­a­tion.

Pre­vi­ous dif­fer­en­ti­able ren­der­ing of SD­Fs did not fully ac­count for vis­ib­il­ity gradi­ents and re­quired the use of mask or sil­hou­ette su­per­vi­sion, or dis­cret­iz­a­tion in­to a tri­angle mesh. In this art­icle, we show how to ex­tend the com­monly used sphere tra­cing al­gorithm so that it ad­di­tion­ally out­puts a re­para­met­er­iz­a­tion that provides the means to com­pute ac­cur­ate shape para­met­er de­riv­at­ives. At a high level, this re­sembles tech­niques for dif­fer­en­ti­able mesh ren­der­ing, though we show that the SDF rep­res­ent­a­tion ad­mits a par­tic­u­larly ef­fi­cient re­para­met­er­iz­a­tion that out­per­forms pri­or work. Our ex­per­i­ments demon­strate the re­con­struc­tion of (syn­thet­ic) ob­jects without com­plex reg­u­lar­iz­a­tion or pri­ors, us­ing only a per-pixel RGB loss.